CHAPTER VIII NUCLEAR REACTIONS
A. dotation
The nuclear reaction A
A(a,p)B
+ a
■3 + p + <4 is symbolized by
alpha, p proton, d deuteron, 3
Pa r t i c ] b s are s jmb o L i z ed by J
gamm a ray , a nd f for fission.
Q is (+) for an "exothermic" reaction, (-) for " ondo thermic" .
ThG thres hold is the minimum energy of the bombarding part-
icle in order far the reaction to occur. Threshold is measured
in the laboratory . system, and therefore is not necessarily equal
in magnitude to Q. If Q is positive, the threshold is, in nrin-
ciple, 0. If Q is negative, and If the bombarded particle A Is
approximately at rest, then ( see ph. I, rage 5)
Threshold energy = (-Q) X Mss_qf incipi e nt par ticle
Reduced mass of svstem
= (-Q)X^i^-
M«
VIII. 1
for the reaction symbolised above,
B. Gen er al Features of Cross- s ecti ons for Mr clear React ions.
The following considerations apply to cross- sections for
nuclear reactions in the absence. of resonances. Resonance phen-
omena are discussed in section D.
Consider the transition A + a— *-B + b + Q, v.here the nuc-
leus "A" and the particle "a" become the nucleus "E" and particle
"b" Both the initial and final states of the system consist of
a pair of unbound particles; therefore the transitions is to one
of a continuous distribution of states. The initial state also
has a continuous range of possible energies, but the experiment
itself specifies a particular Initial energy .
There are sl;r.Ilar situations in atomic physics. For example,
In emission of a photon by an excited aton, the transition is from
a sinr-Ie state to one of a continuum of states:
Conservation of energy selects the final state.
Another atomic example is the non- radia-
tive or Auger transition. An excited atom may
have two: possible modes of decay. In addition
CCtom, no
photon
cttom b dnd-a
photon
to photon emission, the atom may decay by emis-
sion of an electron. Suppose, for example, the
excitation corresponds to one missing electron
in the K shell . The energy made available
vahen an electron falls into this hole may be greater than the
ionization energy, in which case an electron may be emitted from
the atom. Again the final system consists of tv:o unbound particles
having a continuous range of possible energies ,
Returning to the nuclear reaction A + a — *-B + b, we use
a general principle of quantum mechanic s to derive some essen-
tially statistical results on the variation of the cross- section.
From quantum mechanics, the probability per unit time of
141
14 2
transition
"Golden Rule No
Nuclear Reactions
number of transitions per unit time
nil
W =
X
2 «k
Gh. VIII
w is given by
VIII. 2
where H is the matrix element of the perturbation causing the
transition, and dn/dE = energy density of final states, counting
each degenerate state separately.
lXi' a may be the same for all energetically possible final
states; more often it defends on the state. (For instance, \H\
may depend on the direction of emission.) Then \H\* in the form-
ula is a suitable average over the possible final states .****
dn/dE = oo for a continuum
\'Hs\ ^ 0, so that the expression
form x oo . This difficulty is
box of volume JX
of states. But in that case
1M| 2 dn/dE has the indeterminate
removed by limiting space to a
small but" finite and dn/dE large
|M] is then
but finite. JL drops out of the result. The number of final
states equal 3 the number of states of the emitted particle. Thii
is because a change In momentum of one particle compels a change
in momentum of the other, by conservation of linear and angular
momentum of the system.
It was shown in Chanter IV, o. 76 that the number of states
available to a free particle, "b" , with momentum between p
p + dp, confined to a box of volume JTj , is
VIII. 3
and
du =
This must be multi oiled by the multiplicity in the final state**
caused by spin orientation, which is given by the factor (2I b +l)x
(2I B +l), where I b Is the spin of the emitted particle and Ig the
soln of the nucleus. If b is a photon, (2I b +l) Is put equal to
two.*"**
dE = v%
dp^. (true relativistlcally)
VIII .4
where p^ and w, are the momentum and velocity in the center of
mass frame of reference of the final (B+b) state. Since B is
usually massive compared with "b" , p b and v t can usually be meas
ured in the laboratory frame. Combining these two equations:
dn = 4-T^'Jl (2%+lX2T B +0
VIII. 5
From this and VIII. 2 we get
ho. transitions per unit time
LAa|Hl 2 (2V0(2T B + vill.6
Tffi 4 ^
The following equation is essentially
section C^v^g per A nucleus:
definitior
the croso-
* Derived in Schiff, Qu antum Mechanics , p. 193. ("Golden Rule No. l"
is on oare 148 of this text ) .
** This is discussed in greater detail In section C, this chapter.
*** This roint is discussed by Bethe and Placzek, Phys.Rev . J51
450, Appendix, p. 483. Multiplicity is caused by the two possible
independent polarizations,
-&HBH*- See page 214 for more complete discussion.
Ch. VIII
Nuclear Reactions
No. transitions/sec _ Y .,.-- v *f
per "A" nucleus - '** X J "a"rel . to"A" x G A ^B
143
VIII. 7
where A arid. B refer to the (A+a) and (3+b) states respectively,
and ru. Is the density of particles "a". Take n^ to "be l-jfe, cm" 3
(one particle in the volume). Then
J^ Vjrel.to'A" * a-^-b tt^ /K
VI II. 8
Since nucleus "A" is often massive compared to "a", V" a "rel .±o"A"
is often nearly equal. tfl'T a in the center of mass frame. 'In
any case, these two velocity magnitudes are related by a cons-
tant factor. Writing ^"'Vrel .to" A" ~ 1j ~a '
A-?-B TT-K
=Wl^>t| 2 A(2V0t2I B ^)
n^J h
VIII .9
In general, )t is unknown. It has the form J drEfy^ U f in{ ha.l
where U is the interaction energy*. If the wave functions used to
comoute M are normalized in volume JL , Sh. disappears from the
expression \SL H\ in VIII. 9. This is seen as follows: let I have
the form, at large distances, K exp(ikz). Then y"|Yi a o.r = Nlfi
Setting ifjl = 1, we get N = l/<fcfi?
If T, ... -, ard T„ -, now mean the un- normalised Diane wave
xnj_Tri.a_L i ma_L
functions, the matrix element factor In VIII. 9 becomes
sih^ f^T Wl ur lmtal
VIII .10
(This may be looked upon as taking Jt= lJ Henceforth we use ~H
for JlH . In order to show the meaning of this expression, we
write it as
pfs£| = [J X Volume of nucleus X ] T mi tial T-fW)
VIII. 11
Where H|^r in Is a suitable average of the product of the wave func-
tions over the volume of the nucleus. U, and hence the integrand,
is zero outside the nucleus. U = average Interaction energy^
depth of potential well. For our purposes here the Important
feature of VIII. 11. is Its dependence on the charge of the parti-
cipating "particles. If "a", say ; is positively charged, its wave
function will be reduced in amplitude at the nucleus by the
barrier factor exp(-G- a /2), where, by III. 3, p. 5$.
fj^dn.
— c barriers
VIII. H
U & denotes the charge of "a" times the Coulomb potential of "A".
Physically this factor represents Coulomb repulsion. The wave
function of an outgoing particle at the- nucleus is also reduced
by such a barrier factor. The result for the squared matrix ele-
ment is: it— \ 2
For neutral particles: IX I <£(u X Vol. of nucleus) VI 1 1. 13
For + charged oartlcles: |Hi*&(U X Vol .)*X exH-G^-Gj,)
144
Nuclear Reactions
Ob... VIII
femission of negatively charged particles (elect rons)ls treated
in Ch, IV )
For endo thermic reactions there is a threshold energy for the
bombarding particle. For exothermic reactions in which the energy
liberated is much larger than the energy of the bomoarding particle,
there are tm simplifications in equation VIII .9= l) the barrier
factor exc(-G n ) for the outgo lrig particle is almost constant be-
cause it is a function of energy of the emitted particle u ,
which is almost constant; 2) p^ and v b are almost constant and
therefore the statistical weight factor In VIII. 9, Pt/ V a v b* lB
proportional to l/T^f
These results are now apolied to specific cases to deduce the
general features of the vs. energy and d vs. velocity curves.
1) ELASTIC (n,n) (both particles , uncharged)
v.
. ./^ 'therefore" p|/v v, = C*Wt ) > a conBtant
At low energy \H\ Is approximately constant, therefore
constant at low energy.
ELASTIC (n,n)
£.
a le^Y e.v.
0f a
EKOTHEgMl'G. low energy UNCHARGED bombarding particle, as In
(n,.a), ( n , p } , [n , if, ( n , f ) . 4 is usually ,~ Mev «hl le neut-
ron energy is— e.v., therefore v b ^ constant . Therefore
p b Vv a v b Z lAa • r*l E °C exp ( - G n - %1 . exp (- % } is . & c onstant ,
since it den ends on the almost constant energy of the out-
going particle, or, in the case of an uncharged b , is 1
exactly. Also exp(-G r ) = 1. Therefore
a ~ 1/T„ (the "1/v" law)
_L EXOTHERMIC
oj;
3) INELASTIC (n,n' )
The nucleus Is left in an excited state. The process is enao-
thermic and -<J is the excitation energy of the nucleus. 5 or
Ircident neutron energies slightly above the threshold, v e=j
constant, since the fractional change In incident energy Is
small. Put v , changes relatively greatly in this region:
v t (C excess o
n <c
Pn' / T n v n'
f energy above the threshold. Therefore
,. v j i oc ^ energy excess 1 . Therefore near the
threshold cS CC \| energy excess.
INELASTIC iri,v') cr
ENDOTHERM1C
-T We s hj Id
[ ¥ neu TVoN
4 ) ENDO TIIERMI C , CHARGED OUT GO I NG part I c 1 a , as In ( n , a ) , ( n , p )
3) except t^at the factor exp(-Gv,J
Exactly as In case _
operates and is d o m 1 nant . ^ e&pnersy excess 1 X exp(
Ch, VIII
Inverse Processes
145
5) EXOTHERMIC, CHARGED INCOMING particle, as in (p,n), (a,n),
(a.,'3 ), (p, "tf ). For incident energies << Q, the factor
p^ /v a T-^ oC l/v a . The barrier factor exp(-G- a ) operates on
the incoming particle.
EXOTHERMIC
<y cC l/v„ s&cpC-Ga)
In all of the above, no account has been taken of resonance
phenomena .
■ Inve rs e Processes
Consider the transition A + a — >- B + b, where "A" and M B I:
are nuclei and "a" and "b" are, In general, lighter particles.
From equation VIII. 9 the cross- section for this transition is
(neglecting spins): , ,
A _- JffUHr Jt- vt-.,..,
i
A-*^
7T*l 4
rt&^b
The Inverse reaction Is B + b — > A + a.
.2
cr Q
SiH
k'
Its cross section is
VIII. 9"
|uLr*l is the same In both cases, because the operator of the
perturbation is Hermit ian, i.e., I ' : fti*X¥&£\ = \ (w'* U V (tfc\
therefore,
JA^-e
Cs^-A ft*
(neglecting spin) VIII. 14
The same result may be looked at from a different aspect.
Suppose we have a box filled with arbitrary numbers of particles
"A", "a", "B" , "b" . The transitions A + a ^=t B + b occur.
Statistical mechanics asserts that at equilibrium all possible
states of the system consistent with the specification of the
energy of the system are occupied with equal probability. If a
state consisting of a pair of particles ~A + a is called an "A"
state, and similarly for "B" state, then the occuoied states in
the energy range AE may be divided into the two' tyoes , A and B.
Since all states in AE are equally probably occupied, this div-
ision Is such that
No . occupied A states
No,
Is-
No.
possible A states in A E
15"
occupied B states No. possible B states in AE
The number of possible A states = maximum number of (A + a) pairs
times the number of states In A E for one oair - 7? 4"rr fa. _&
%
At
where '( - maximum number of (A + a) pairs formable With the
particular numbers of particles cut into the box Initially. Sim-
ilarly, the number of possible B states
where J} is the same. Therefore
71
146
Ho
No. of
Inverse Processes
of o ccupied A states __ Pi ^b
c cue led B states
Pfc>a
Ch. VIII
VIII .16
mw at equilibrium the number of transitions A-^B equals the
number of transitions in reverse, per unit time.
No. transitions A-^B/sec = (Ko
No. transitions B^-A/sec = (No
Combining VIII. 16 with VIII. 17,
A states occupied ) A ^ g ^
B states occupied )CT e ^ fir to
VIII, 17
VIII. 14'
iSWB'Uv
t^a
as "before
If the particles have spins, the density of state 3 is in
creased
If" the spins are I A , I a , I B , I b , the density of A
states is increased by the factor (2I A +1 ) (2I a +l ) , and similarly
for 3 str-t.pp Thfin t/hft rate of transition A— i-B is proportional
to
"tes. Then the rate of transition
(2I A+ l)(2I a+ l)p a 2 C^ B
and
-A to
<2i 3 +i) (2i- 3+ i) Pb 2 cr e ^ A
therefore
(2VH2V
, <U B = (2^+0(2^0^^
VIII. IS
Note that in this formula, O is an average over the various kinds
(spin orientations) of A states, and a sum of partial Cf s for
various possible final states. *
* This may be elucidated by writing O^g more explicitly. Divide
C Into contributions Ots) due to various relative orientations
of I. and I n . The number of states represented by each relative
orientation is 2Sf 1 , where S = resultant angular momentum of part-
icles "A" and "a". In this discussion, orbital angular momentum
Is neglected. It is Included in a discussion in the appendix of
Bethe and Placzek 1 s paper. Phys.Rev . SJL 450. The total number of
A states Is (2T A +l) (2I..+I } . The total cross section for transition
to any B state is
%-
E(2 s ^)tf(S)
average over spin states. (S takes on 2I a +l values
the partial cross
if t a <' J i A V "(2T A +i) if"i A <i a .) Nov a\s)
section- for various initial values of S, nay be written as a 3um
of contributions to various possible final spin states, I.e.,
C*sl =X ^% ,
where i denotes a particular final spin
state of the B + b system. &&\ contains in addition to the dens-
ity of states in ener~.y, the squared matrix element for the parti-
cular transition represented by cnSTf . For transitions not
con-
serving total vector angular momentum , 0(5).
0.
For examele ,
Ch. ¥111
Compound Nucleus
Problem
to ■ g i
Be^ + H 1
147
Design an experiment to detect the inverse reaction"
Li u + He"
(Design of the aloha particle source will depend on the thresh-
old energy for the inverse reaction. From Allison, Shapes ard
Smith, Biy a .Rev. 21 550, or from Hornyak and Lauritsen, Rev"
Mod. Fry s. SO, 202, we find that Q for the forward react iorTis
2.115 Mev. In the reverse reaction, in order to get 2.115 Mov
into the center of mass coordinate system we must give the alpha
an energy of about 3-5 Mev, and this is the threshold for the"
inverse reaction (See section A). Design of the Li6 target and
of the detector, and determining the required aloha beam strength
require knowing the cross section. This is got by detail balan-
cing arguments from d^^^ , taking into account a spin fac-
n° r o° f m 7 ^ - ™ ThiS cr ? ss section is f ound in Livingston and Bethe,
'-. rev.Mod.Hiys^ 9 245, p. 310, or in the original source, Allen,
-, ^...^u.iny^ y ^-j p. jiu , or in the original s.
Ki.ys.Rev ^1 182 Tl937), and is 5 x 10^9 cm 2 " at 0>1
Mev
The
cross section for the inverse reaction Increases raoidly 'as^the
volume of phase space available to the proton is increased
therefore it Is advantageous to use alpha energies an Mev or
more above the threshold of 3.5 Mev. Higher energy orotons also
penetrate the Coulomb barrier readily, and are easier to detect
A qualitative curve of cross- section for the forward reaction
as^unction g^^is given in Hornyak and lauritsen, Rev.
D. The Compound Nucleus
""in the diagrams of section B it was assumed the lH^was
approximately constant, except for the Coulomb barrier factor
Often, perhaps in most cases, the matrix element has irregular
variations. This phenomenon is called resonance . For ^xamol-
if [n J S J Process 1» indium, there is an extremely pronounced
peak in rf at a neutron energy of 1.44 e.v. d reaches 27,000
barns at this energy. (one barn is in '' '" ' "
this resonance peak is 0.042 e.v, = p
curve of rf vs. energy has the form
is the resonance at E R - 5.2 e.v. rf
for the (n,# ) reaction in silver.
In this case d reaches 24,000
barns, and the peak has a half-
-24
em .;
Near
l7(E-E R } 2 .
The half-width of
the resonance, the
Another example
will
H - 0.063
e.v
consider the reaction n
for a, 0; assume for A,
initial spin states :
tial spin s bates for
= 4; for S
B
3,
The
3/2.
+ A — >C +
±; and for _.
^2(l)+l)(S(l/2)+l) = 6.
total angular momentum 3
1/2, (2(l/2}+l) =2.
*)s = i < *%-f ) +
Now Hie first term represents
states having S = 3/g. For a
spin3 are, for n, 1/2;
The total number of
The number of ini-
= 3/2 is (2(;5/2}+lJ
^Ai%
transitions to any of the final spin
given initial orientation, there is
only one. Similarly, the second term represents transitions to
any Imal state having total angular momentum 1/2. hut dnce the
spin of the a = there are none, so 0T(&) =* 0. Mien orbital ang-
ular momentum is involved, there may be more than one way in which
tne given Initial state can form a final state, so that &&) for
tf^lll' I 3 a T °^er the various possibilities. See BethTand
Plaezek, Pays. Rev^ bl 450, appendix.
148 Compound Nucleus Ch. VIII
The explanation of this phenomenon lg baaed on the assump-
tion that the transition A + a— hB + b occurs through an inter-
mediate state C:
A + a — *- C —- 3 + b
State C Is the "Compound nucleus". The idea of the compound, nuc-
leus is due to Bohr.*
The idea of how resonances in cross section result from this
assumption can be obtained from the quantum mechanics of second
order tnansitions. The probability of transition, per unit time,
is given by "Golden Rule No . l" :**
trans, urob./sec —
4\
^*X
BC
2 /energy ■
x density or
\staies
VIII. 19
provided there are no direct transitions from A
section is, from VIII. 9,
, ' ...:-K
which becomes , analogously,
&n\
i
v*v b
1
HcA^cl'-^
"A^B-t^ I Ea-^
% v b
:o B , The cross
VIII .9'
VIII. 20
Near E^ = E c , (resonance), tS is large. This formula gives infin-
ite <5 at the resonance energy, hut the formula doeH not take into
account the short lifetime of the compound state. A correct
formula is derived In section F.
The life -time of the compound state is long enough for the
nucleus C to "forget" how It was formed ,***and this results in a
basic simplification in the interpretation.
From the Heisenberg relation AtAE & }4 , the lifetime of the
compound nucleus and the uncertainty y in Its energy are related
P > JL
lifetime
VIII. 21
The reasons why the compound nucleus has a. lifetime greater
than zero are the following:
1) For charged particle decay, the barrier factor (VIII.12)
reduces the rate of decay,
2) Decay by X radiation is very slow compared to the times
in wnich the nucleus changes Its organization: the lifetime
against V emission is ~ 10~13 - 10*3.4 sec. The characteristic
time of the nucleus, I.e., the time for a iiucleon to cross the
nucleus, is —(size)/(velocity) vug 10-2.3/109, or about 1CT 22 sec.
3) A particularly important reason is the tendency toward
equlpartltion of energy in the nucleus. The excess energy due to
the absorption of the bombarding particle is distributed among
all the nucleona. It Is rare that there is a fluctuation in
which a large fraction of the excess energy is on one nucleoli.
4) Selection rules forbid some modes "of decay.
* Bohr, Nature 137 344 (1936)
** Schiff, n. 1967 eq. (29.20)
a-ture 137
. P- 1967
**'* Discussed in Peierl's review article In Reports on the Progress
In Physios VIII (1940), Fhys. Soc . of London, 1941.
C ompound ETucl eu B
149
c-
Ln-
tie
ted
1
r
)
s
to
:res3
Ch. VIII
S. Examrle of an Unstable Nucle us
" An examnl c of a nucleus which plays the role of an intermed-
iate-state compound- nucleus for several well known nuclear reac-
tions Is Be°.
The ground state Be decays
Be^
2 He
as follows:
■* 110 Xey r »
The reaction is barely exothermic. The G9i*W exponent for
decay into a's is low due to low nuclear charge,
see equation VIII. 57, o. \62 . The theoretical estimate of the
lifetime Is ICT^sec".** , corresponding to a width of between
1 and TOO e.v. This time is long compared with the nuclear
characteristic time of 10" 22 B ec . J hence the width of the level
is small.
Information on the excited levels of Ee S can be obtained
from studv of those nuclear reactions for which Beo is the inter-
mediate compound nucleus state, such as Li' (p , ~y )Be e — j^2a,
Li7{p J n)Be7. These reactions are discussed here. The energy
levels are plotted in FIG. VIII. 1.
l) a-a scattering. For two Coulomb centers, the total scat-
tering cross section is oo . ¥e may study the scattering at some
anr-le'"not near (90° in center of mass system is best). We
expect peaks in the value of CS when the incident relative energy
equals the energy of excitation of an excited state. For a s
scattered on a's", the first such resonance should come at 0.110
Mev (in center of mass system), corresponding to the Be° ground
state. This resonance is presumably Ten sharp, a few e.v. wide,
as mentioned above. It has never been observed experimentally.
Problem, Discuss the possibility of experimentally observing
the resonance expected in alpha-helium scattering at an energy
corresponding to the Be 6 ground state, i.e., 0.110 Mev In the
center of mass frame.
(The Coulomb barrier keeps alphas of this energy at least 5 x
10" 12 cm apart classically, so the effect of nuclear forces is
probably undetectable. Also the experiment is difficult because
the range of 200 Kev alohas is so short that it is hard to shoot
them through an appreciable number of scattering centers and
detect them. Any' attempt to detect a resonance might- be guided
"by the experimental procedure of Devons ( Proc .Roy .Soc . A 172
127 and 559 (1939)), who Investigated alpha-helium scattering
at higher energies. The theory of a-a scattering and its rela-
tion to the Be^ nucleus is given In Wheeler, Phys.Rev. 5j? 16
and 27, (l94l) J
A second resonance, this one experimentally observed, Is
at ^3 Kev. The barrier factor is lower at 3 Mev, hence the
state has shorter lifetime and greater width. The half- width
is estimated to be 0.8 Mev.
Further resonances In a-a scattering are so hroac as to be
scarcely recognizable as resonances. All the resonances men-
tioned so far" correspond to states of even parity . This is be-
cause a's obey Bose-Einstein statistics and have symmetric wave
Hemmendinger; auoted in Seaborg and Perlman table
Rev. Mo d . Phys ^ 20 585-
Wheeler, Phy s . Rev . 59 27-
Isotopes ,
Compound Nucleus
150 Compound Nucleus ch . VIII
functions , * The incident a ' s Mil hav e ango lar rsomentuiB 024
with respect to a target a particle. Therefore states of Be*
detectable by a sea Iter Inc. IjS helium are even states.
Not all states of Be 8 are ever;. Odd states of Be 8 c-mot
decay directly into two a's or into the even Be 8 states mentioned
Emission- of electromagnetic radiation must occur first,
" : - i -- -'■ "hange to an ®J| n state by "mech-
Char.ge of parity
above ,
because
an even state c ™t
anlcal" , i.e., r:on- radiative, interactions
occurs in emission of photons.
2) bl (p,K)Be . There is a nrominent and narrow resonance
at a proton enerrrj of 440 Kev . This Indicates that the life-
time of the excited Be is long and thus that it is an odd
state. It decays through the relatively slow process of ~W
emission to the much lower even Be states. The energy of a
Ll 7 and a proton separated and at rest is 17.2 Mev higher than
that of the ground state of Be 8 . Tbe Jf 's given off in decay
from the excited Be b state produced in the IA?fot, ^ ) neaction
hav e ener gi e s of 17.5 Mev and as. 14 . 5 Mev , indi eating tf decoy
to uhe two even states mentioned in the oaragraph on a- a scatter-
ing.
That the excited Be 8 state oroduced in Ll?(p V) is oM
accords with the follov.ring considerations. The most probable
ease is for the bombarding proton to be in a high S state which
is even. The hV is odd, as is suggested by the arguments in the
following paragraph. Then Li 7 (odd) + oroton (S state) is an odd
Be state.
The picture of a nucleus as built up of "shells" of ^rotors
and neutrons, somewhat like atomic electron shells, su.g-ests that
Id' in the ground state is odd.
Suppose the average potential for the nucleors Is a square
•jell The single particle approximate quantum mechanical solut-
ion to the problem loads to orbits which ^ay be designated' Is
lp. Id, etc The Is orbit accomodates 2 neutrons (soins opoosed)
and two protons (spins opposed); ip accomodates 6 neutrons ana 6
pi'Otons , etc. ?
yH
would have the configuration;
protons: Is 2 lpl
neutrons : 1 s 2 ipp
o, o
or a total configuration: (l8 2 lp; ls 2 lr 2 ) , which has ±\l\=-% and is
nence an odd state. This model of the nucleus is discussed in
_ / change s
'■does not
x -^-x,y
■5'3tl
IS
Chan;
y ,Zr-V -
z, then the ?
* For two identical particles, parity of the state and symmetry
functions are simply related: If the wave function"'
ign when space is inverted by the operation
ias( odd oaritv. If ? chan Ges
^even ' does not change
hen the two particles are interchanged in eosition, then T is
antisymmetrical
symmetrical
Operation of inversion of space: Y(r-,,r ) — *■ t 1 (-n ,-rvJ
Operation of particle interchange :f r!'r,) — *- + flro «T
But for identical particles, r-^ i^rfL ^ s^fi
^ £lf-^J - T(r 2 ,r-j), and inversion is equivalent to particle
interchange.
CH.YITI
Be Com|=ound Nucleus
151
Be"
tU
IB. 5+
7 pwi
-oIlO
Zoo
19. ',J
IS. 1 3
17.6
I
in
' ' i".
2.
In
R 8
be
pis.Ynr.
4)0 *)
2.22
Cu ft e s give cross
section for the partic-
ular reaction.
Numbers l), etc. refer
to subsections of
section E .
(Based on "Energy Levels
of Light HucleV' bj
Hornyak and Lauritsen.
Rev .Mod .Phys. 20, 191 )
O of energy
(Energy in Me v.)
Be c
152
greater detail in section
7*
Ch. VIII
Compound Nucleus
in section K.
3) Lt 7 (p»p' )Li 7 *. This is similar to 2) except that a
r rot on is emitted having less energy "than the incident proton,
leaving Li? in an excited state. The resonance in C is observed
at a oro'ton energy of —1.05 Mev . Li'* decays by emit tine a °
of "bout 0.45 Kev. The 0.45 Mev splitting bet ■-.-sen this excitea
state and the -round state Li? may be due to enermy difference
between Pl /n and py states of Ll 7 , on the nuclear shell model:
SS.T3 Mev.
<■
U t7
17.21
,45"3Mev
X
Be*
Cem^»>»4 Nucleus
^r^^Msta*! tsf eder-g/^ \ Mev
Li 7 + p(l Key) — Be —
4
lf£nevg/ = O at,
of Be s ,
gvoun
d stat
Li
+
Li
V
+ P'
4) Li 7 (p,a)He
> 7 i
md
odd, the incident nroton must be in an odd. state,
state with respect to the Li 7 nucleus. Ko reson-
Since two a's are in an even state,
since Li 1
probably a p' ...
ahc'es are observed. None is to be expected, since all even
resonances are extremely broad. Note that all the observed
re sonance s 7 are odd^level s , ,„__,+. __ D .^„„ r .„
5) Li'(o,n)Bn'. A resonance is observed at a proton energy
of— 2.22 Mev, corresponding to an odd Be b state 19.15 Mev above
the ground state . ,-
6) Li 6 (d,a)He 4 No resonances. Evidently Li D + & form an
even, state, and quickly decay to two a's.
Problem: Design an experiment to observe the famous 440 Key
resonance in the reaction Li? + p— ^Be" -+/+ Be 1 - ->2a ■
(This exoeriment has beer performed by ha Ike r and McDaniel^
using a gamma ray spectrometer which measures the energy of
pair's orodueed by the gamma ray ( Phys.Rev . ^4 515) and by
Delsasso, Fowler" and Laurit sen ( Phys.Rev. 5JL (1937)) using
a cloud chamber to detect the gammas by means of electrons pro-
duced in the chamber by pair production and Gompton collisions.
Recent electrostatic accelerators have been equipped with
electrostatic velocity selectors which provide an energy spread
in the motor, beam of less than 300 e.v. at 1 Mev. In order to
take advantage of this narrow energy range, very thin targets
must be used. These oroblems are discussed in 'Gamma-Radiation
from Excited States of Light Nuclei" by Fooler, LaOTitaen and
Lauritsen, Rev. Mod. Hays. 20 236 (1948)).
F. Quantitative Development of Resonance Theor y; Brelt-Y.'i.-.ner
Formula . t ^ \
In this discussion we use as an example the In, ) reaction,
1 . c . , rad 1 a t i v e c a otu r e of neu fc.rone , wh i ch is a n 1 mpo rt a nt re;:, c -
tion.
As in the preceding sections, the energy levels of the ini-
tial and final states form a continuous distribution. The exoer-
ment picks out the particular initial state.
^y ie resonance phenomenon that we wish to describe is very
enerry -sensitive. Ve shall attribute it to the existence of:
compound rue "Lens state C, connecting with the Initial and fir
states by matrix elements "H Bt . and Mfat , where the notation is
'i-nn.l
e
Bj
Ch. VIII Resonance Theory 155
given in FIG. VIII .2 We shall neglect matrix elements of the form
H & q> j Mty&tiA M a y , which connect the initial state with itself,
the final state with itself, and the initial state with the final
state without the intermediate G state; i.e., no direct transitions
from A to B,
STM"E A CornswJD Nucleus C STATE B
A+/n C B *t
inde* a.
— — index b
(discrete) (fort-hrwous}
PIG. VI II. 2
The problem of computing transition probabilities can be
done by using the complete machinery of time dependent pertur-
bation theory-. In summary, if u* are the time independent elgen-
functlons of the unperturbed states having energies E. , the true
eigenfunction for the perturbed state is
Y =1 % A,
J,
E,t
VIII. 22
where a 1 are the amplitudes of the unperturbed states In this
expansion. For the method to be useful, the true eigenfunctlon
must not differ greatly from one of the unperturbed eigenf unctions .
If there Is no perturbation, a n = for all n. With the oertur-
bation,^ , the a's change according to the equation
a =
^Z H^
cue
s(w^*
K
-A*
Hjt^dx.
VIII. 23
If j just before the perturbation is applied, the svstem Is in a
state represented by uv, i.e., a, = t- k , then some 'time after-
ward there Is some probability or finding the svstem in states
other than the k zh . For states differing in energy greatly from
state nnm.ber k, the exponential in the above equation oscillates
rapidly and the change In a ji tends to average to 6.
to the
, of B
Applied
states by eL
E„ by d c (see
state Is near enough to be important
the one- level Breit-V.'igner formula.)
the amplitudes are "
for
present problem, we denote amplitudes of A
states by a-v, , and of the compound nucleus at
rd ■:/'- ill. 2) . Assume that only one intermediate
(The result is theft called
The differential equations
a.
= "^K
q, = -^H tc e
ife-Ee)t
Sx
1 a.
fc-t
The
lnlt:
_al A state ,
say
ct n
= 1,
a„ = for a
£ a
J*
a,
-^ZHcbe
M^-Ztf
Is
b
chO£
en
exoerlment .
VIII. 24
At t
It
I
m, Ch. VIII
Resonance Theory
154 - ■ n h,rt a ^ a increases, representing
At t = a a = a. = but a ^ O .a intermediate
the build up of probability that tne eys airplitu des a c and
State C. When a c becomes larger tnan , the *»pJ-l de £ ot9B
a b begin to Increase. Tne a^ 1 *^^"*^ a^ituds that
A + ^ ri^ous ,ay to solve -r the transition P^abllltles *
is to carry through the solution of the system oi
^^ollO^E 1. a practical way to K et the answer quickly.
Assume for the sa.e of «*^^ t t*^ ^tn^
compound nucleus state is ^^^^^^Son^rooabillties are
tegrate either to A or to B. The transit^ P ^ ^ ^
^Sa b Ii;SiS B o?S^ta?e SSst'a certain mode of decay.
Probability of transition to B - ^ - £ [Ate
(per sec . )
= i4^4 c3 VIII. 25
where we have put ^/c = p y , v y - B ■
Probability of transition to A = ^r -^l^vi *
(oer sec . )
(Trc Wave functions are realised in a volume Jl = 1.) The decay
from the state C ivould folio,; the eauation;
Prob. of occuoation of state G - g
i'(t^)
VIII. 27
Actually the probability of occupation of a states ^am P litude| 2 .
.If we ignore the phase factor, _vrejwrite_
amplitude = NJprobability
t/ I ±A_\
or
satment
5 4 — (
VIII. 28
Kere thorou^ treatment confirms the result of this step.
Defining
VIII. 29
VI 1 1. 30
n ^P + P -fe/a) times the total probability of ^structlon
I rrU+ U -^% % ^l intermediate state, per unit tune.
( r is sometimes defined ^h^T^T^ZlA^ ce^ritSn
probability of decay p er uoit -.lme.J me.- vjl-l-u- __ .
* Breit and W^r, BHfeBg* g g9 Clg6)
** Froir now on, Q c is not ae^essaxxiv e.quaj
Ch, VIII
Re s o na rxe Th eo r v
' v 155
We muBt add a tenm to account lor the accretion
J*., e - —*■ ""** s* l.^-x:u uu account lor the pncrp+in
of the state G. State C is IiM frcm a+ .^ . h ' „ C f^f
appro-n^iate +pt«, fv,™ -,-. t L IT irom state A accordin- to the
ViilJJ, rametf £* " S ati0r ;r e °^ **»*«* to equation
Therefore
we try
VIII. 31
SmK^SitV^S S^TS^f eqUati ° n 3atlsf nn G the initial
a, ==
^H^te^-^-e* 1 ?
VIII. 32
tJsl nr fho ,,-„ ■*-•„■ experimental
wid?L! h t\rSiSS£? ?i^ C Sr S t^ °^ /tol ^e Of the energy
turns out usually to be ?esf ThS 1 gSfg ™ 3 St f e G ' ^ ■ „
seconds the term exr>( Vi /%) IT ^ C * Arter ^™ i^ 13
Jn e;x i'\-\ V)iJ is nearly 0. Therefore
VIII. 33
P'+fec-EO'
ri2. 8t Sen^ lldS ^ ^ ^ am ° Unt re3?rea9n ted b ? **• «1« of
Number of reactions to the ri^t/sec
B + y }
= Ucl "^r
Number of reactions to th= L6£t/fie*
(A + n^G-^A + n 1 ) " =
L X
VIII. 34
T'
LTi,
examine" k] ^"it^SarS^^T^, ^ ln the flml fo ™> «
c „ J ' cl early Ilas the form of a resonance curv^-
By definition, T aaJ T are fcrj* ■•'-..-.
connected by 2^ = ^. Slne:# "
I I turns out to be approx-
imately the uncertainty* in
ener G y of the state, this "is
an aspect of the Heisenber-
uncertainty relation AETS'tf.
"Energy /v.dt h eft
%_ maximum ^ O f -1
Returnin G to equations VIII .34, ;,rit!n. ; them explicit!
Wo. ncactions/seo foWl*
A^G^E (per = fT~ . —
unit beam density) ' ^v t e~-£«J
No. reactions/sec |>t co '*
'^'-^^
unTt bTam density) ^'+ (£t- E g J a fcjj 7 ~ *W)^<ft
Viii. 35
where the definition of Cf 1 r equation VTTT 7 i„
assume the r,robl«n jq r„,™*n , q \ , lon ' m '7 is used, and we
cf neutrons"= lT -iodized to volume^ = 1 and density
156 Resonance Theory Ch. VIII
(in deriving this result more carefully, one should substitute
a c into expressions for a a and a, , then put these g, ' s and a^' s
into the correction expression f on a , and check that the result
is consistent.") , ,-;
J _ ^ ] ^ Ca > {
M ^j p, + { ^ T VIII. 36
21; i>u„r
The next step is to replace \%J with more useful parameters. We
use equation VIII. 26 which nay he written
M 2 "" ""'"
m«
* = i¥?T 3 1 ^ ! v
H
VIII. 26'
:f | ra denotes U for v„ = velocity of neutron if it had the right
energy to hit the resonance exactly = v R , then
~i^)H*J e M'
VIII .37
provided 1Hcq,| varies little with change In velocity. Rewriting
VIII. 36 in terms of the parameters v R and ["*- ,
°fm,*l "-
C%/n) —
2g 1 2ir^rJ
VIII. 38
VIII. 39
M**S
Wow use the relation ^ =i U, K Cv/v R ) , where v is, as always, the
actual velocity of the neutron. Also,^ ^ tf/Mv , | ^ ^ /Mv R .
M = mass of nou.tron .
rvr R
VIII. 41
v = actual velocity, v R = velocity of neutron to hit resonance
exactly. .The aensral nature of VI I I. 40 and VI I I. 41 is shown in
FIG. VIII. 3. Note that for
energies "below the resonance ,
the eqvntioas above behave as
described in section B, p. (44,
So far, angular mom°ntnm
multiplicity has been neglected.
For the .{v., IS ) reaction the spins
are: feutron, I H = l/2
Initial Nucleus, I A
Compound State Nucleus, I c
-' J C^ih/h)
F1G.VIU.3
CJli VI11 Resonance Theory
Final Nucleus, I B
Photon of fixed momentum has two degrees ef
freedom and ia lifce a uarticle of sein l/2
(see ref., footnote *#», p,|^j '
---^j* Sl °'" r r" eutr,0n tactions the neutron usually has no on-it-i
DJmTl X G - % i 1/2 resonate. The cross sec-
tion must be multiplied by the probability that, -or giver, I. am
are^Tf^^/S ^ ' *** "^^ ^ the ri " ht ***»t*tl<m. There
- T Wo V + * initlal St " tSS havl ^ ; t0tal an C^r momentum
- I A ^ + 1/2, ( 3pln . a parallel), and 2(l A - i/o ) + x st ,, tR3 ^
total arxcular momentum = I A - i/ 2 , ( api , s ant inarain el ) . Ihfi ""
total number of states Is a<l + i/ 2 } +1 + 2(l A -i) + i = 4l + ^
The probability that an ineomlm neutw™ ,diT ^ f ■
state uith the nuclei-. j^ mlIJ S n "' ltl011 ■'HI f™ a scin-parallel
4r a + 2
21, + 1
VI II. 42
and that the neutron forms a state with spins anti-parallel is
j^I A -/ a ) + [ _ T A
4I A + Z
23* +
VI II. 43
V: ' '' " - l be ^oniblned, ~ivin S for the final equations;
VI 1 1. 44
^)-^U,-^^s
Cfyyft,} — 4-TT A K
£J&
*r
v s +^y
21* +
VIII. 45
where the + is used if I Q - r^ + w £ - ar)d _
There is little or no resonance if b i t
neutrons. C F ^
if I
+
1/2 , for
- 1/2
slow
nomena
in mo-°o 1 S 4r + 10n CU ^ eS r ° r £ ey -*^ ^bsorntior and scatWrinr
il . , "Science and Eiipi-
m^j^ 19259 fl.947) or in Goodm
neerm;;; of luelear Power", Vol. I.
^ m „ At the ^ddle of the periodic table, A = i 00 to 1 50 fn ¥ 1
aroc esses are ^rmil npn+ n-p+ D v, «„ -. " -o<J , in, C )
for /Sta is ~ fSr tL^ pGrl f iC f^' the M ^0 energy
Shis amount At +m * h ' L compound, nucleus is excited by about
closed s^c^+niL ? r r °f ^^ation, the levels must" be
VIII 4 3 ht t ° n " U ? 5? r ^ c -- 0Sel 7 spaced resonances, FIG
viii ^. But at Iovj excitations vre find that the ener-r levels
***** Cr ° 3S ^^^^^^^^u^Ti^rcso process^
158
Experimental Resonance Data
Ch. VIII
are of the order of 0.1 Mev aoart.* Evidently the— Is a -reat
S^ Se ^^ 1SVel dens "y wit* increase in excitation
onei {^y .
The increase In level density has been lucidl^ eXDlaiaed by
Boar. Suppose that the nucleus resembles somewhat a collection
of narmonic oscillators of different frequencies «* . Eacl has
energy gc**^ (ignoring zero point energy ). The total energy is
E =1W^ W ^*-~^u^ . For low OMltatior _ f fBW Qf the
Kj differ from 0, and the energy channea in iumns of ~&*J which
we may take as -v, 0.1 Kev . For large excitation, say 10? e!v
many rjj s are largo; there are in general manv sets of m*a that
£l> a J^S 1 e r- Gr ^ near t0 a lar Ce value of excitation Each
such set of n t 's will in general giy. e a slightly different total
energy, therefore the density of levels will be large. This result
clearly depends on having many nucleons. Ve expect few energy
levels and few low energy resonances in the light elements the
number of resonances Increasing with A. This expectation, is fnl-
lll.lea to some extent, because there are few loiv'e-iem (<1000
e . v . ) r e s o na n c e s In el erne nt s lighter than mangane s e . ^ ' "
+ n ( F S r , vo ^ hi #i atomic number, the density of levels accessible
to [n^ ) processes does not increase; altho hi~h«r A means ior P
degrees of freedom and denser levels, «**«« more
the binding energy of the neutron
decreases from ~ 8 Kev to 5 or 6
Mev, and thus the states for for-
mation of compound nuclei by absorp-
tion of a neutron are at lower
excitation energy and the density
of levels is correspondingly less.
These opposing trends tend to can-
cel .
Not all levels are detectable
by (r.,^ ) processes. Those for
which the spin of the compound
nucleus state is not equal to
■^A ~ 1//2 ' a "l/oi" parity Is not
conserved, do not give rise to res-
onances in { n> X ) cross section.
Throughout the periodic table
I Y is roughly constant at a value
0.1 e.v. v.dth variation by a factor
of 10 either way.
For medium and heavy elements, V n s= lcr 3
ponds to a life time against neutron decay oi ■
Is large compared to the nuclear oeriods of about HT"**- sec- the
captured neutron moves around — Iff? times before a neutron escsnes
ihis very low probability of escape can be understood by consider-"
lng partition of energy among the many nucleons. (See n". 14 & ) this
process to make neutron emission slow operates bettor as- A becomes
larger.
In light nuclei, VL. is larger than in heavy nuclei, as ore
expects from the partition of energy argument. " \Z ler-e means
large probability of scattering comoared to cantnre. In mangan-
ese^ for example, the resonance at 200 e.v. is" almost pure scat-
Energy
Levels of
Compound
Nucleus
FIG. VIII. 4
c o rre s-
e . v . Shi i
10~ J - seconds, this
-21
* Known from gamma emission spectra and from comolex
alpha spectna.
lM e v
c; the
escapes ,
orsfder-
}. ) Thl s
becomes
Ch. VIII
Nuclear Gas Model
159
out. 1% as larpe as the
VI I I. 44
teri-'i" * Trio rjeslc in absorption is only .
scattering peak. The Breit-Ki -ner formulas VI II. 44 ano 05 are
fitted to this resonance by putting b> &*1 o.v. ana V^ 10 B.-.V..
In TP rv UPht elements, there are no (n, 5 } re sor-.ar.ee s at
ell. There are, however, (n,p) and (n,a) resonances.
The Breit-Wi-ner theory may be extended to reactions involv-
ifin clia^fed rarticles by ircludinp the barrier factor clue bo
Coulomb" forces.** For encrpies above the hei-ht of the Coulomb
barrier resonances are observed. For example, the reaction
''■' ^ H? has a 'uroad resonance at 0.27 Kev.
Li 6 + r_
■He
Problem. Assuming that U»\y, find the possible values for
C for the 280 e.v. resonar.ee in manganese. . CJ - , , \
( i at 280 e.v. is 2.73 x 10" -1 cm. The spin of hn JJ is 5/2 J
Froi? enuation VIII. 45 f
At resonance k = 0. Then ipnorinp Vy compared to k , we pet
If v;c hnov: that P^lOO hV , ve can also find %,19.
5
Stati sti cal G-as Model of the Nucleus
Va r i ou s m od el s o f the nu c 1 eu s empiia size various^ different
features of the nucleus. Me si nple simple f" -}}
nuclear properties. V;e shall consider tne statistical oi ^f
model, then' the liquid drop model apolied to ussion, anc finally
the nuclear shell model,***
Tne -as nodel nic tares the nucleus as a C as of protons and
neutrons/' This model ignore, surface effects Itte na .piliaritv,
a serious omission. The volume of the pas Is W ,5 ; "^
Due to this restriction to a small vo u,e ^^ILsTv^r/Sfih
mm for s rertide has widely spaced levels.*--" Dffles.s very p^gn
e^ciStlen^nercies are postulated , P^I^a will ? c W toel jx-
_.« + w911 »i,i 6 states. For the usual nuclear excitation encr.ieb
oflQ lev or so! the nucleon ras Is almost completely aepenerate .
We shall use the m model to compute an approximate nuclear
.etentiarSii'depth, !o explain semi^antitatlvely the increase
in nuclear level density uitn ener G y, and to consider smisBltti
particles as an evaporation process.
The excitation of the nucleus, the ^tert to .chich hipiher
T - cerr-sS? to complete degeneracy . At T = tne number
L-%?aKa ri to the hiphest one occupied just equals tne nun ner
or reticles, either Z or A-Z, depending on uhether the proton or
neutron part of the &s is in question.
71 6
'.;.
* Seidl, Karris and Lanpsdorf, Phvs.Rev
#* See Bethc- B, Ch. XIII, pape 186 . „ fl ... fa1/1
*#* Various nuclear models are discussed by L. Rosonleiu,
Nuclear Forces I I, n. 185 , and by Seine B, p. (J
**** Equation VIII -5, "for enerpy density of states, cortains
the volume .
160
Nuclear Gas Model
Ch. VIII
The number, n, of states of momentum less than r> ,„ of a
proton confined to a voiumeJQ, is
r ' = 2 ■ — ; rvj (factor 2 is for snln, see p. 142}
3(2ti40 3
At complete degeneracy n — Z, therefore
^bX^A
* f
%Z ■-■■ (? v )^k&
4
VIII. 46
Similarly
VIII ,47
In the crude approximation, number of neutrons — number of protons
=N=A/2,
4L - (3Tl) * inm
A
= 1.05" x l& cgs umts
0.5*^A 7 3 ' : VIII < 48
;S!: = :S independent of A. The corresponding kinetic energy is «v 21
Mev ^y^a . ^Mf, Is the kinetic energy of the highest occupied
neutron state. This energy is measured from the bottom of the
potential well. Further more,yW is about 8 Mev below 0, a 3 shown
in FIG-. VIII. 5 This fixes the depth of the well for neutrons.
Actually there are fewer protons than neutrons in the nucleus, but
the topmost proton level must have energy^^o , otherwise there
would be 3 decay. Therefore the depth of the well for protons is
somewhat less than for neutrons, by this model. Also, the Coulomb
potential acts on protons .
Ipl-oton |ootenilQ I v.
Oeotron potential
8 Mev
~ 21 Mev
B&."ES,S"
or
The following, is a calculation,,! corresponding to a usual
value of nuclear excitation. From the statistical mechanics of
a degenerate Fermi-Dirac neutron gas.* the total energy of excita-
tion (measured above the T = level) is, for neutrons
(FIG. VIU.&)
# Mayer and Mayer., Statistical Mechanics , p. 374. The total energy
measured from the of kinetic energy is
1 ; + ""J VIII. 5P
VIII. 49
E*-*»*xL'+1F(jS3"+
<m. viii
Nuclear Gas Model
161
How put In Tor / M 6 = p^ ax /2M , and then for p put VI II. 47. The
result is
Similarly
x-:
ftfejA h&^s
rV^^^-^^f) 2
172/
#
_n-
fc.
p = fW ^ f^f f/rf!
VIII. 51
VIII. 52
where ^ =nuclear volume, T = nuclear temperature, and M = mass
of a nucleon. The total (neutron plus proton) excitation energy.
/
FIG. VIII. 6
T
E t - E n
+ E
P'
is
X,
(Me/)
VIII. 53
Where E^ and (kT) are in Mev. If
A = lOO^and Z - 44 , HL = 11 (kT) 2 .
Then (kT) = 1. Kev corresponds to
Ex— 11 Mev. If we attribute to each
kT a "decree of freedom" in analogy
to classical statistics, then in this
case there are only 11 decrees of
freedom, whereas there would be 3A
degrees of freedom in the classical
E£ s .
The increase of energy level density with energy of excitation
can be computed using the statistical-mechanical definition of
entropy:
Entropy = S(T) = k(ln w(T) - In w(0)) VIII. 54
where w is the total number of quantum states available to the
system at the specified temperature. Fro is thermodynamics,
str)= f T ^ » r T 2aVTdT = Zo ^J sine* E-a^feT}' VIII. 55
a a
So 3 =^ ^fl^^ r
From the definition of entropy,
,7%. _ _ US (Tj _ dens ity of states at temo,
uj ( 0) ~ density of states at temp.
e
VIII. 56
e = e
11 (kT)^,
For A = 100 and Z — 44 (same example as above), E
and at E ~ Mev, the excitation of a compound nucleus after
capturing'?" neutron, e ^ ^ e l9 ~ po 8 . Therefore, if near excita-
tion the level spacing, is ~ 100 Kev , at 8 Mev excitation it is
~ 1 millivolt, by'this'very approximate calculation. This spac-
ing is probably much too small, although one expects this calcu-
lation to give' a greater level density than observed by resonances
because resonance experiments detect only levels with spin compat-
ijgg Nuclear Evaporation ch ■ "Hi
ible with those of the initial particles.
Nuclear evaporation . The emission of a neutron from a nucleus
may bo considered an evaporation of a particle from a statistical
group of particles held in a potential well.* In such an evapor-
ation, the particle carries away an energy of order of magnitude
ST, which is, in general, much less than the total excitation
energy of the group of particles.**-, For example, suppose a nuc-
leus'" of A = 100, Z"= 44 has excitation energy, £=20 Mev. The bind-
ing energy for one neutron is about 8 Mev. The temperature will
be about, ""l. 3 Mev. A neutron will, on the average, have kinetic
energy of about 2.6 Mev after escaping. Therefore the nucleus is
left With an excitation of about 10 Mev., which is sufficient to
emit another neutron. After the excitation is reduced below the
binding energy of a neutron, the nucleus may decay dj gamma omi-s-
si on.
Protons encounter the Coulomb barrier, which has a height of
^j .gr-10 Mev for medium weight elements. The probability that 2 +
~ 5 Mev is concentrated on one proton is small if the total ex-
citation energy is of the order of 20 Mev, therefore protons are
less likely to' escape. At small Z, the Coulomb barrier is rela-
tivelv small, and p emission processes compete with n emission
processes. For largo Z, n emission dominates.
When the energy delivered to a nucleus is very
than the binding energy of a particle, as in nuclei
c o sm i c ray part 1 c 1 e s , a ' s a nd ev en larger nu cl ea r
"evaporated". In photographic emulsion or in a cloud chamber.
a "star", having 3 to — 20 or so prongs, is observed r (See
much larger
excited by
frarj-ients are
17?)
Problem
Plot the probable number of neutrons that an excited
nucleus will emit, as a function of excitation energy. The
plot should take the form: probable
Plot up to energies such
that four neutrons have
probably evaporated . Discuss
the probability for emission
of a proton at this excitation,
nurvber <£
neutrons
Emitted
Assume A = 130, Z - 54, and that the binding energy of a ncutron
or of a proton is fl Mev .
(The temperature of the nucleus changes only a few percent when
a neutron leaves, so the error is not great in assum'ng that T
is constant. Gamma decay may be assumed negligible for energies
of excitation "of over a few Kgv above 8 Mev (see Bethe B,
p. 160). Considering emission of mavtnons first: For the energy
region 0-8 Mev, no neutron can lie emitter]. For the region
{3+n, f sv: Kev)to 16 Met'-, one neutron is emitted-. For 16-24 Mev
there are pb.mpeting modes of decay, namely, l) one neutron may
take cnoumh energv'to prohibit further evaporation, 2) two neut-
state population
met ion for the nuc-
Energy, E b
ro"-s may leave. The
lcar mas has the f orm :
We may approximate the
tail of this function b
an ezepo .ential, and say
that t.-~e A rrobabillty for
escape = C-v/u.-tt'e'^' 1 '; £>Efe
where C is a norrmlisatlo::
factor and £. is energy measured atove^, , the Fermi energy, th e
Frenkel , Phys .Zo it, s .Sow .1etu.nl on 9 533 (1936) (in English)
■- Accord in" to Weisskopf, Phys.Eev. ^2 295, the average energy
of evaporated neutron is about 2kT .
energy
Ch. VIII
Nuclear Evaporation
163
energy of the highest occupied state at T - Let i = E - 16 ■
^, 1G the e xclt -tion ener^. Then the probability that oilfe neutron
takes out so much energy that leas than 8 Mcv is left ii " ' ""
> - */VT
vanes
c" d et e rsai ned
8tS
3
8 + 5
c At
die
= c £, kT where
setting P, =r 1 at 6
0; C"
p l =
- p -feT
Probable number evaporated, excitation 8-16 M°v = P + °P =
P L + £(1-P 1 ) =.2 - e"^ T 1
For total, excitation 24-32 Mev , there are three eomnetinr
process: 1) one neutron loaves not enough energy for further
evaporation. P-, = e"^ 8 *»% calculation^'slmilar" to above . This
is neplipibl.e. 2) the first neutron evaoorated leaves enough
energy for just one further emitted neutron. The nrobabllttv
for t'nis is i-tt+s .£. _ S - - - ./
p ? - G l e FT i £ = e *
|) the. first, neutron leaves enourh enerpy for two more neutrons,
ana either one or tv:o more are omitted. The probability that
case 3 J occurs vrith just one neutron beinp evaporated subsequen-
tly can he shown to be _„ g / - 1- \
p 2 = .Ff(, e ' V- Then the Probability
for ease 3 J with- ftwo subsequent neutrons, i.e., three altogether
is ■ j ■
% = 1. - Po = 1 - H - P3 = 1 -
The average
= 3 - e
number for
'<>+*)
the 24-32 Mev rann.e is
The plot turns out as -iven:
For protons. the barrier is
effectively higher, by an
amount that Is mnroximatoly
equal to 0,9 times the peck"
height of the Coulomb poten-
tial. T> Le probability for
penetration of a Coulomb barr
by a charged particle is
P = gl^PT^^'K^
Efret^y
infi W)=i
O-sy
where x = Enerry/Barrier h&i^hi
(after Bethe, B*, p. 167)
A, and 2 1 are mass and. charge of particle, Z ? is charge giving
rise to the barrier. A 2 , the atomic weip-ht, enters through the
nuclear r ? ,n;, n . ft -. ™ + „ • Curves of probab-
nuclear radius, R^. For a proton, A-.Z-, = 1
ility of oenetration for three valui
given:
v\
\\ A =
90
\A 2 -a
2 \
X \
A, for protons, are
rVotnri rVstnstod/eftcy
.7
'Eme rgy
« = Barrier
Fission
Oh. VIII
164
The ratio of probabilities for emission of a neutron and. of a
proton is ._. ->■ ?wev
C-o
on
-cm")
-4e
LtffMtV
~ .00 37
where 9 Mev - 0.9 of the 10 Mev Gculomb barrier. 1.65 is kT
c o r re sp o nd ing to an excitation of 38 Kev , av e ra ge numb e r of
neutrons emitted = 4.
h
Fission **
The most useful model for explaining fission phenomena Is
the liquid drop model (see chapter 1,0, p. <S ). This model per-
mits calculation of the change in potential energy when the nuc-
lear drop suffers an ellipsoidal deformation from spherical shape.
If the potential energy Increases, spherical shape is a stable
configuration.
The two contributions to the potential energy are l)
capillary energy, 2) electrostatic energy. We wl.ll calculate
the change in these contributions to the potential energy for
a constant volume prolate ellipsoidal deformation given by the
equations
ma lor semi-axis = a = R0+ e )
H"
minor semi- axis
b = R/\fi
1 +£
O— <3P> VI11 - 58
where R = Initial radius, £ = parameter giving extent of deform-
ation. (3-S- approaches the square of the eccentricity of the
elliptical section as both arra roach 0.) Volume is invariant:
V = "(W3)ab 2 _ (4n/3)R3.
l) The capillary energy is proportional to the surface area.
Ellipsoidal surface = 4tt P 2 ( 1 + 2/5 £*+...) VIII. 59
The capillary energy was c omp.it ed In Ch . I , p . 7 , and found to
,;De 0.014 A^ for an unexcited (spherical) nucleus, therefore
capillary
merry = 0.014 A 3 (l + 2/5 E*+...) (mass units)
VIII. 60
-, n a
2) The electrostatic energy = (3/5) fe-Z^/R) (l - 1/5 %hk &t
sphericity the energy is, from Ch . I , p . S , 0.00062 7 Z-/A,
therefore
Electrostatic energy - 0.000627(Z'-/jml - l/g E ) VIH.61
This is evidently maximum at spherici"
The total change is
£ (2/5 x 0.014 A 3 - 1/5 0.000627 &'/$%)
VIII.6J
Spherical shaoe is stable if this is +; unstable, if -. Roughly,
■electrical energy is proportional to A* 3 , capillary energy to A ^,
Therefore the electrostatic energy term dominates at high A. The
expression VIII, 62 gives a criterion for stability for given Z
and A, namely,
Cvt-'i -
Spherical nuclear drop is stable if t "C4.7 VIII. 63
* Bohr and Wheeler, Phys.Rev. 56 426 (1939); Frenhel , J.oi
Physics, A^ad.S c l.U.S.S.R. Vol. 1 Jo.2 (1939) (in English)
**Gencral reference: W-.E .Stephen, "Nuclear Fission and Atomic
Pov, r er," Lancaster, 1949.
Oh. VIII Fission
The plot, of Z 2 ,/A shov:s that elements up to U 2 3S
stable by t]iis criterion; they are In reality.
Although spherical shape
of the nuclear drop may be a
relative minimum of potential -2
energy (i.e., metastasis), the ^
potential energy may be even
louer for completely" separated
halves of the drop. To investi-
gate this, v:e can compute the
enerry of tuo separated spherical
drops of equal volume having a
total volume equal to that of the
combined sphere, and compare this
ulth the energy of the original
d ro p . The ma s s f o rmu la 1.8,
p, 7 , may be used for this ■■ur-
pose. If M(Z,A) denotes the mass
are Interested in the difference
165
at least, are
50-
*■!>
— - -
-^
40-
3S-
3<-sS
20-
to-
/2S
D .
Z=o
2o
4o
SO SO I DO
120
of combined nucleus, ther: ue
M(A,Z)
HV 2'1 J
%
0.014 A 5 (l-2 3 ) + 0.000627
S*
(i
This gives the
volurr. e fragment a
Ccrence In energy I or
At, closer distances,
Infinitely separated half-
still not" touching each
In FIG. VIII. 7, the
other, the 'potential increases due to Coulomb repulsion. These
facts permit d raving an approximate potential energy curve. For
y236 the separated halves have energy -169 Mev relative to the
combined non- excited nucleus, FIG. VI II. 7.
early part of the curve
represents deformation
of the sphere into a
prolate ellipsoid of
small eccentricity, and
Is known from equation
Fill .62; it la quadratic
in the parameter £ . For
Fully seoarated fragments ,
" ~ no"'
the curve Is
I) r
it to r c , the separation
lny^erbolcLj
CoulomD betential
Distance j mengm^
ith E. at sb^Vi t d^Tov-^dtiDn
distance at which the tv.'o
half-volume fragments just
touch, the potential dev-
iates from the Coulomb lau
due to the onset of nuclear
forces. (in the case of
-,f- ^Volume Spheres
\S- J0 , if the Coulomb lav; just touching pi S .y[if. f 7
held right un to r , and If the fragments remained strictly spher-
ical up "to this point, the potential energy would be 210 Mev J
The c v rv e in the re pi on near r is c ompl 1 c a t ed , Cal c : : 1 a t i o n s o f
potential energy for large deformations (up to about point B on
FIG. VI 1 1. 7) of the nucleus have been performed by Fran!-: el and
Metropolis ( Phus.Rev. 72 914 (194?)) using the "Enlac" computer.
Due to aero point energy characteristic of
systems, the system has energy slightly above t"
potential well .
There is some chance that
undergo " snor-tar.eous" fission
the nucleus will
:y tunneling
qu a ntu m- m e c ha 1 , n I c a 1
■ e bottom of the
fig, vm,7
166
Fission
Ch. VIII
through the barrier. The probability is lovr, because the masses
are large and the system approaches being classical so far as
tunneling through barriers is concerned. Therefore, effectively,
there is a threshold excitation for fission.
In 1939 Bohr ana V.heeler* deduced a rule that has been
verified experimentally. It is that the reaction (n,f) 1b pro-
duced In elements haying an odd number of neutrons oy thermal
neutrons, but with elements having an even number of neutrons,
fission Is induced only by high energy neutrons. This is because
changing from an odd number of neutrons to an even number of neut-
rons releases one or two Mev . This rude agrees with the table
of which is reeroduced here. The
part
Possible Bl'vichviq £^e.v
guen
aid
N
Q
e
o
- 7
~ 5"
~5~
given In Ch. I, p. 7
rule Is exemplified by
fr 35 and U 238 . The bind-
ing energy of a neutron
to the former may be some-
thing like 7 Kev; to the
1 a 1 1 er , wh i c h already has
an ev e n numb e r of neutrons ,
only about 5 Mev. Fission
induced by thermal neut-
rons occurs in Ij^3?5 o U t
not in 02^8 _ Xli tnis case, the barrier to fission is presumably
around 6 Mev. in height.
Asymmetry of Fission Fr , _
to split into equal fragments
Ti
sent distribution. Nuclei tend not
Fragments tend to cluster in two
zones, as shown in FIG-. VIII .8 and 9'. Referring to JFIG-.VIII .3 ,
the fragments, after formation,
move toward the line of max-
imum stability hy p decay.
There is no adequate theory
for this a symme t ry .
Extensive data or nuc-
lear species produced in fis-
sion appeared In Rev ,yod . P^s ,
18 513" (1946).
fission fragments
tend to be in these
regions, rather than
midway .
Stable nuclei
— i
FIG-. VIII, 8
Fission
Yield
1
fo
I
lo" 1
/^il
/ 140 N^
f\
I
\
&o
to
1 0<j 1£B
l4o
L4&
ltt.-A
Kejnarhs on Fission Fragments . Fission fragments nrroduce heavy
cloud chamber tracks about 2 cm. long in air at NTP. * In contrast
to ionization produced by a particles, the density of ionization
here decreases with d 1 st anee t rav e 1 1 eel , as sho we i n FIG- . V 1 1 1 . 10 .
The explanation is that the fragment , Initially to a large extent
stripped of electrons, gains electrons as it slows down. Its
* Bohr and bheeler, Rhys. Rev. 56 426 (1939)
eh. viii
Nuclear Orbit Model
167
■■fission fragment
About 1%
effective Z decreases, and ionization depends on Z (equation
11.10, 1. *bh (The a picks up char G e also, of course, but the
effect Is smaller.)
Emission of Neutrons in Fission. Immediately after fission,
when the f ramnent" Tr drops' r are some-
what like: OQ ' each fi*^>»ttb
uossessss considerable excitation
energy. This energy la used to a
large, extent to evaporate neutrons
There is approximately ono neutron
produced per fragment
Delaved Neutrons
of the neutrons emitted by fission
fragments are emitted at relatively
Ion": times after fission, I.e.,
from 55 seconds to a fraction of
a second. The explanation Is as^
f n m n vrq in the "B decay by which. n
f fsio">ragn,ents\ecome stable nuclei, It ^JJWJJ.fjJ § + 1
aeeav from a neutron- rich nucleus 2 to the ground state ot-4 + 1
f'SrMdden. Then the nucleus Z+l is PJ^^^ft 11 ^ ,
U-+-* n - n fl n aY if It has sufficient energy, transiu^u v '
^f by^vSorating a" neutron. The neutron binding energy Is
relatively small for these neutron- rich nuclei. It can, ol
processes leading to the nucleus (Z+1J . aenDLeB e^v
Triple Fission. Fission of a large nucleus Into three
fragments--- two major fragments plus one alpha particle— Is
known.
Distance
FIG. VIII. 10
Problem. Draw quantitatively as well as possible the curve
FIG VIII .7, the energy of a nucleus as a function of some para-
meter giving the extent of deformation. Invent a suitable
nnwunp+PT to measure distortion. „^„„„
f It is probably simplest to describe the deformation by surface
harmonics, I.e. , p _ a + 2> n S n
1 + ■ „ vM + in^ raflltis and Sl, are surface harmonics; for
where a is the initial radius ana ^ invarlance Is thus auto-
SfcaliroroJldeg rorT Good Approximation to experiment Is
obtained by Sling r = a + b 9 P 2 (cos6). The capillary energy is
-ents are not joined, but are still close together, tney are
deformed something like: QO The energy can be .rudely
approximated by assuming each to be an ellipsoid The papers
of Bohr and feeler, l,c . , Mel and Met nop oils 1 .c . «* ,
of Frenkel, J. of Phys. Aca d. Scl .USSR 1, So. 2 (1939) Un *n S _i^- J
are pertinent to this problem. ^
E
rb 1 1 Model of the Nucleus
H^gi 9 |SiSr^ nucleus In terms of nucleon orbits
^J nv°» the ascription of the atom in terms of electron
Sits, rnrorbi/olcture'ls valid If collisions are — -ough
so that a nucleon may travel at least across the nuel •££**£*
col lision. This requirement seems at first not to dc -u±±-l.l_
h;-As far as terms linear in the br
J n
168
In a v
ucleus ,
Tor
is of
the ord
er o
leons,
the saean f
the nucleus .
How
ignore
s , and
th e s
1
) When
one
Nuclear Orbit Model
Ch. VIII
at ?** 20 Kev the n-p scattering cross- sect ion
f 0.3 barn, and, for the known density of nuc-
ree path is only about l/j or so the radius of
ever, there are two factors which this calculation
e make the orbit picture appear not so untenable,
ential well. If
Spaced hue leons ,
blend together to
2 ) The uucle
energy states are
occur between nuc
ferring both the
principle prevent
state . Diagramma
nuc 1 eon passes
the nucleon is
the wells may
form a roughl
us is a degene
, for the most
leons only if
nucleons to em
s two nucleons
tlcally ,
another, It passes through a pot-
constantly passing other closely
be so closely spaced so as to
j uniform potential.
rate system in which the lowest
part," filled, A collision can
the collision results in trans-
pty states. The Paul! exclusion
of the same kind in the same
Initial momenta
of two nucleons
plotted In mo
entum space
approximate be
few, but some
collision cause
direction. Bot
must represent
states. If the
states on the 1
lision does not
if occupied sta
completely fill
there could be
all.
mdary of filled
unfilled states
_, to change
re 1
ew end points
prev 1 on sly empty
re are no empty
ocus A, the col-
occur. Thus,
tes described a
ed perfect sphere,
no collisions at
state region;
here
Neither of these tuo Ideas
extent that they represent
is justified.
The orbit model has been explored
absence of accurate information, a squ
with rounded corners, is adooted as th
The depth of the potential is assumed
A. This Is justified by the computati
showing that the kinetic energy of the
^A*a , is, to fair approximation, inde
the gas model. The requirement that t
neutron be about 8 Mev then fixes the
las been fully investigated . To the
-.he true situation, the orbit model
ith some success. In the
are well, or a square well
e form of the potential,
not to change much with
on at equation VIII. 48
highest occupied state,
pendent of A, according to
he binding energy for a
total depth of the well.
Quantum mechanical calculations for the square well give
levels which may be denoted as follov. r s :
Is Ip Id 2s If 2p lg 2d 3s Ih 2f 3p 11 2g 3d 4s
6 10 2 14- £ \1 id 2 2 2 (4 6: 2£> IS l° Z
2.
2
4D
TO
\\z
16 8
The letter gives. X in the usual way, that Is,
means ^ ~ ~5 , etc. The number gives the nu:
s
r o.
state
capacity
summed no .
of occupants
means^r = , f
radial nodes.
"l" means no node, "2" means one node, etc. (This differs from
atomic spectra notation,
n - J - 1. )
:ere the number of radial nodes is
For a sharp cornered well , the states listed are about equal!]
spaced In energy. When the corners Eire rounded off, the states
shift so as to clump into the groups of states given in the list
Ch. VIII Nuclear Orbit Model 169
above*,* for example, Id shifts closer to 2s.
As mentiored above, the depth of the well 1b approximately
constant throughout the periodic table The radius R increases
With A according to R = 1.5 X 10~H A /3 . For small radius, only
the Is states are hound. As the radius Increases, hound states
of higher It become possible, in the order listed above.
The Paull exclusion principle prescribes the maximum number
of one" kind of uarticle that. can occupy a particular space state.
just as in electron configuration of atoms. There may be 2(2^+1)
identical earticles in the state having angular momentum A . The
factor 2 represents the two possible spin orientations.
We may attempt to build up the species of nucleus out of
nuclear shells, just as the periodic table is obtained in atomic
theory:
P resumed configuration (P;K)
il H 2 D (Is- Is) meaning a nroton in Is, a neutron in Is. Could
' ' ' have spin or 1 and still accord with Pauli
nrir.cinle. Actually observed spin 1.
2) He 3 (Is 2 ; is) Spin 1/3 necessanily ;
must be opposed.
since the proton spins
(lsjls"-) Spin l/2 necessarily, similar to above.
(is 2 ; Is 2 ) Spin 0, a "closed shell" nucleus.
3) lh>
4.) He
The is orbit for both neutrons and protons is now filled,
The next orbit, lp, accomodates six neutrons and six protons,
Giving 1 " as the next closed shell nncleus.
5) He 5
does not exist; evidently in He 5 , (is ;ls"lp), the
It? state is not bound." Hq5 is the compound nucleus
* This may be understood by the following argument . Consiaer a
sharp cornered square well having ei gen state functions Y ± : lhe
perturbation In energy of state k when the corners of the well are
rounded can be found "from first order perturbation theory, ««■«£
J- as the unperturbed states,
shift in energy is
The
■potent to. 1 .
= cJj4^M^>
.c = fir,
J2
joe-rtuvb'.ng"
H
where ix* is the radial part oif-vf* ,
H Is the perturbation of the poten
tial that occurs when the corners
are rounded . Let H = H-, + &* . H
1
gives the rounding on the inside, at.
A, H 9 gives the rounding outside,
i p r m at_B. „ j&
H 2 Is negative, but c| Kl^ > C L \AA*& ] therefore AE is +.
Now the reason that states cf higher angular momentum have larger
AE's Is that particles in these states spend relatively more
time near the edge of the well, in region A.
170
Nuclear Orbit Model
Ch. VIII
for the reaction n + He 4 — *-jj + He 4 . A scattering
resonance Is observed at about 1 Mev , presumably
representing the lp level of KrP . In fact, it 'is
The sb
claimed that two peaks are observed.
p l/2 and %/g
would be the p., /n and p^ ^ states;
an He
(Hall and Koonta, Ph
. Rev , 7k
6) ti l
i 2 2
(la ,lp-;ls ,lp)
presumably
_E!SrgV
of n
; 196)
The end
this "period" is
There are two independent systems.,
each consisting of a soln 1 (orbital)
and of a spin l/2 (intrinsic spin).
There are many possible total angu-
lar momenta, including the observed,
namely l .
16
the next closed shell
31! 3.
It has spin and is unusually stable
On the orbit model as given so far, the closed spoils for
either the proton or neutron confl durations are at 2, 8, 20, 40,
70, 112, 166,.... neutrons and/or protons. For l&vr.e nuclei the
Coulomb energy makes the number of proteins less than' the number
of neutrons, and the nucleus cannot have closed shells of both at
the same time .
VIII. 64
of neutrons,
Empirically, the closing numbers appear to
2, 8, 20, 50, 82, 126 (Magic numbers)
Nuclei having either such a number of protons, c
are unusually stable.
These are not the same as the closed shell numbers given bv
the preceding development. This discrepancy has been Interpreted
by M. a. Mayer as follows. Suppose that spin-orbit coupling splits
the energy levels corresponding to different J values, that is,
lg splits into lSoy 2 and l®^'. Assume that the level with larger
J is mono stable, i.e. lies lower. This assumption is not contrary
to any known facts about the nucleus. Then the former closed shell
number 40, for example, must be alt ©red as follows:
Old dividing line
between shells
(Higher i numbers are
a a erased to have larger
Bplittmg)
5~0 ]nuc|eans
u|» to here
Hew dividing line between shells
M.
d. Mayer, Phys.Rev . 7_4 235 (1948)
Ch. VIII Radiative Capture H(n,"tf )D
The next shells are
1&7/2 2^ 3s lh,
171
T.l/2 1
82 nucleons to here
lh g/2 2£ 3P lt 13 / 2 ,
126 nucleons to here
Thus the altered nuclear shell theory Gives closed shell numbers
that anree with the numbers deduced from experiment.
Problem. Look up the nuclei having; closed shells of eitner
neutrons on protons, and note to what extent they tend to toe
relatively stable .
L. Capture of Slov.- Keutrons by Hydro gen
This nu clean reaction is one of the few that can be calcu-
lated with some precision.
Only the S wave comoonent of the resolution of the neutron
plane wave into radial functions is important for the low neutron
enerrj.es to be considered here, i.e., thermal energies, ^ > O
waves have angular m omen turn with respect to the proton so great
as to keen a slov: neutron beyond the reach of nu clean forces.
(See chanter VI, section c, , p.ue-rtj The only source of angu-
lar momentum is intrinsic spin. The final state Is the deuteron
3^ state (the 1 S state Is not bound).
n(slow) + H ; — >• D + a ( itiw =
The conceivable transitions are
1
.19 Mev,
"S (continuum)
■^S (continuum)
( magnetic dipo'Ls)^ 3g
■ ->- 3 S
(is not possible,
a s shown just
after eq. VI 1 1. 78)
The parity selection rale excludes electric dipole for S-> S
transitions, Ch. V, p. loo .
The physical mechanism,
in classical terms, is the
following. Since the magnetic
moments of the proton and neut-
ron are not equal , ^p ~P J^h j
the totaly^f for the deuteron
does not have the sane direc-
tion as the annular momentum
vector, FIG. VI II. 11. There-
fore the .system is a rotating
magnetic dipole. The formula
for magnetic dipole radiation
is almost identical to that for
electric dipole radiation,
differing only In that the
magnetic "moment matrix element
Is used (see Ch . V, p. 95 )
Momenta
2.7
Moment* j
^ Y-W
resultant magnetic
moment moves around
the fixed I axis
FIG-. VIII. 11
178
Ch. VIII
Radiative Capture H(n, t )D
Probability of transition ^ ^ \ u, \ t,,,~r ,--
7 unit time =^X^S l/NWl.Jwfad I VIII. 65
oer
where W = (energy of transition)/^.. The energy of the transition
is 2.2.3 Mev, the binding energy of the deuteron, provided the
energy of the incident neutron is sraall .
The wave functions .are
[final) T(%) =
(++)
., j( (+-) + (-+) )l/j? s z
s a = + i
(initial) % } s ^V( + _)^-+3]% yiIIt66
Notation for the spin functions: (+-) siear.s proton spin up,(+)j
neutron spin down, (- ) . The space parts of the wave functions
are independent of angle, being S functions.
u(r)/r is the solution of the Schrodinger equation for the
known potential, U-* , the triplet S state neutron-proton potential
The radial equation is
M-U) + T^(^)(\W 3 -U 3 !^)) J W =
VIII. 67
W-, = -2.23 Mev; U4 - -21 Mev for r smaller than the range of nuc-
lear forces, and beyond. (See theory of the deuteron in Ch.vi ,
p. us .) u(r) has the form:
u(r) c£ © ' for r>R. k* - \ YmV#J - 2.26 x 10 12 cm" 1
** -hit
The error committed in using u(r) = G g 3 for all r is tolerable,
since R < l/k v and we shall correct partly; as follows. When
u(r) is set — C, 6" 5 , we get somewhat too large values in integ-
rations over r. If we compute the normalization C using the same
approximation, an error is made which tends to compensate. Normal-
izing so that , f * ..j
fir = 4TC Me 5 | d& = 1
we get
So
VIII .68
VI II. 69
VIII .70
j(r) is the S wave component of the resolution of a plane
wave Q-~* s into polar eigenf unctions, but perturbed by the poten-
tial well of the proton, The potential well is U-, , the potential
existing between a neutron and a proton in a sir. "let S state.
Q* & is already normalized to one particle per u""it volume in
the absence of the 'oertnrbinr. ootential well, The unperturbed
plane wave can be exuanded as follows*
&K IM^)/&)"LN Pit— )
VIII .71
*3tratton, "Electromagnetic Theory," p. 408
Oh. VIII
The S wave Is
Radiative Capture H(n, T )D
WfcS,
-) 4N
JkJt>
173
VIII. 72
We must now find how U perturbs this. The m mature of the
wave function is areat+y incr eased in the region of the well
and is nearly Independent of neutron enor-y, "since V-c neutron
energy is small compared to the well de^th". From i'ib theory of
scattering of neutrons by nrotons, Ch . VI, m Itnow t'^t for si ok
neutrons the tangent to the radial function ) (r) at the noter,-
tial well ed-e V, intersects the axis at a distance a-, from r -
Therefore ;.{r) must have the form:
W =j(x)
i=*r-
tje^mntn^ o"f tne function — -
in fe
vm.73
It Is evident franj the fi-are that j ( r )« (l/h) sin(kr + ia.,0
a fairly good approximation to the tru
for small r.
Computation of the matrix element
t *
! fimt
j (r) . j {r}^ JajJ + r
The matrix element is
VIII. 74-
The marnetic moment vector is
VI I I. 7?
where C T etc . are the Paull spin operators for proton and neutror
These operators act as follows:
<X x (+)^(~)
a(+)^i(-) cr„(+) — (+)
i 6
VIII. 7G
For example, C^ ( + _ ) _ („) f le£vlnr> thG lzeutTOn sr , ir , B ymtol u^-
cna n red . We c al c a 1 a t e (jj^) a s a n ex ampl e .
For this we need J^<|^ +ja n CT 71 ) , the spin parts of which -ive
r nw — /
rr^
3 ^ ^ / \7.
Therefore
(Note t hat if Jj F - ^J N the matrix el em ent would be 0. H>- sleally
this corresponds to the case that the resultant ma -net! c moment
has tne same direction as the an^lar momentum, Which is invariable
1V4 ' — -— Ba&latlye Capture H(n, 5 )D Ch. VIII
and the average magnetic moment would be constant,)
The term above has the same form as the spin function for the
final 3 Z - state, and is multiplied by
C++)
3-*-M
i> = i^ [c+-)+£-fn^ s,-o
Due to orthogonality of the spin functions, only the ¥
S z = gives a contribution. Therefore
final
for
reduces
Therefore
VIII. 78
It can be seen now that -^S (continuum)
''S (bour.d) does
not occur, for in such a transition the matrix element depends
on the integral over two S wave functions for the same potential,
and this is 0, by orthogonality of eigenfunctions of the same
Hamiltonian. In the transition considered above, 1 S — 1-^3, the
3 functions are for different potential functions.
Since there is no preferred axis, \(df.J)^\ 2 =^f./h\ l ~\^li)iJ\
\M U V=3Wi,)S TOI - 79
from VIII. 65,
.From the definition of the cross section & , a
we get
°C^r)
i#r-rAd V ji*}Mfifa
VIII .80
where a factor l/4 comes from the fact that only l/4 of the ini-
tial continuum states of the neutron and proton are singlet S.
A
Using the values: I13 = 2.26 x 10
12
OL
a l
CA K 10
2.32 x 10" 12
cm
cm
W
rus
VI II. 81
VIII .82
§W velocity (f%j
For thermal neutrons, V = 2 ..2 x 10 5 cm/ sec, 0^*} = 0.29 barn.
The experimental value is O.30 barn. An elaborate theory includ-
ing tensor forces gives 0,31 barn.
Remark s on neutr on canture in 1 i ght nuclei. C^caVM is in the
range .001- .01 barn, with some exceptions,
H, with c^
feaj^tj
= »%
the
Ch. VIII
Photonuclear Reactions
175
jvL°£2 f + them - p is rela tivei y meb a ls due to fche fact that
:!.:^ dCU !'f ro r- State is alcost bo1 -^, i.e., because I aj is
exceptionally large
light elements, but
. There are other cases of large.-*** in the
toy the reaction ( n , pa rt i c i e ) :
B^O C( n ,a) - 3800 barns
He?, 0"(ii,p) = 6000
Li
f.3 cr( a , a
-*. , - !«.»«, i - 8000
W 14 , C(n,p) = 1.7
+ ?\f 0r fI ' J? SfS'll, compared to the others, because, in contest
factor ISf^f i ? 8M * 8 ' + V 8 larG6 en ° U ^ t0 Eake *S barrief
iactor ejcpt-g-j » important for emission of a charged uarti-cle
IftEfSSer'or g*.**"^ ° f * ^on is reduced^ fgcSr"
M. Fhotonuclear Reac tions
The Be are :nainiy"rY7n), althou^ U.2r_), (J.fj, etc are
hnown also. The threshold for (jr.n? is the binding energy If
«°° n ^ USUal ? ab ° Ut , 8 Mev " ^eptions te this figra are
■■■"■-■■ : thresholds of 2.2 and I.71 Mev, respectively
Strong J absorption resonances have been observed
at 30 Mev ( t h re sh o 1 d 18 . 7 Mev } ,
m
,Gu
2^^l; at 22 Key, and in „^Ta
at 16 Mev, Goldhaber and Teller *» have interpreted these res-
onances as due to an electric dipole interaction in ,M '\b the
neutrons." Btt ° 1 * lB mm a ' S a Unit with ro ^ ect ^ aS the
Phot gd i s 1 nt e ,-rs.t io n of the de utero r
pliotonuclear reaction having a reasonably
The transition if + D — *-n + p (unbound.)
This is the only
reasonably detailed theory.
Except at
1)
2)
lov
J S
3 Q
(33 .ground
(D ground
fttto
p (unbound) occurs in two ways:
3G^r.ej -*- b I continuum)
sl c»n e ) -^ s- ^p [ ooati nuum )
er.erraes
tenths of an Mev above
gular
to be
- - ---, —.Av^tjiwo-g \Li_.j to a fe 1 .-: tenuns or an
threshold), the latter is the dominant reaction. The
momentum of the P state requires too high a velocity
attained by lov,- energy neutrons.
rp , p + ^ fjr* m °f' 3 S— ^(continuum), is the inverse to the
reaction discussed in the ore ceding section. Therefor© its
cross section is obtainable from detailed balance arguments,
allied e #*£ be va^VnTJ^ 6aUatl ° n VIII - Bl ls ^ st *&»*'
iUocl t0 bs valid ^or ail neutron energies. The result is
S .U - Vi* ^ ^ = H' EK + W '
f,(j +W.)
VIII.83
where Cu D is tbe angular frecuercv at threshold, namrdv
finding energy W, and Wo is the magnitude of the fictitious
binding, energy of the deuteron sin -let state **** " " "
* For definition of & see VII 1, 12 or I II. 3, o 58
^ Goldhaber and Teller, Phvs.Rcv. 74 1046 fl948).
**"* Bethe , D , p . 58 " "
***•* W Q is really a parameter such that
state were bound, its energy would have to be W
tne observed singlet scattering cross section,
FIGURE in this footnote, which is continued on
if the singlet deuteron
in order to
Shown on ti-
the next page
p
jivo
176
Photouuclear Reactions
Gh, VIII
The cross section for the second tvne of transition 3s-»-3p
can he computed usiri" the formula for G amma ray c^sorptiiii.:*
CT^^^lMp VIII. 84
^a la the reduced mass, ano v the velocitv of the emitted
particle. M is the matrix element for the electric dipale moment
M -sr«* i »'^8*
VIII .85
£»« ^ coordinate of the proton with resect to the cento- of
ST+H^ 6 + faCt ? r tf? rssulta because the neutron is uncharged,
ano tiie system is lihe half a dipole .
Wis will not carry out the calculation, taut merely describe
the approximate wave functions.
Except when hi rh est accuracy is wanted, T, 3 „. may he featoan
(fro m e qua 1 1 o u VI 1 1 . 7 ) . In t-h e absence of the umlear got ©Btlal
*(3p) is ths P wave component of a plane wave (see equation
VI II. 71), and has the form I {$*** ^i^ ^^ + , „
.SaITCl &&-#*-} iron tne properties
^ x ^ a -^-°Et d Vessel functions.** At loir energies, (A) ir FIG.
vi_i.it;, Tp p j is nearly in the reel on of the short range
nuclear potential well-, and Is therefore little disturbed by
it, una is like tne P wave of a free particle. At hlrdier ener-
gies ¥ ( 3 p ) lias shorter period, and is Perturbed m 0re by the wall
(B) la FIG. VIII.12.
*fe
PIG. VIII.12
r¥ is plotted in FIG. VIII.12 because the behavior of I is seen
best when matrices are written in the form, ^cc Unf*) f (vif;) ch.
The result is: ^
U CS» f 7 3 M cOj d J Uj 2 - j VIII .86
where UJ 6 is the frequency at threshold, as before. The two cross
sections e^Oiwfctffc) and ^^(etetii^are plotted In FIG. VIII. 13
fro**.?.
^ "
actual
* Pethe D, ■
** Schiff, n".
— a,
. 56
77
fictitious Uond state T^ _^
Wing same Qi j nud eneig/ Wo.
Very High Energy Reactions
177
e\ectvic
of If
The reason that Cfy^fw^) rises so sliarply is that energy above
the threshold (0 excitation) is almost a resonance energy. Ac
can be seen from equation VIII .83,, if W q were zero, there would
be a resonance at excitation energy.
g. Remarks or Very Hirfi Enerr.Y Phenomena
- a) Stars . A rroninent feature of very high energy phenom-
ena is the production of stars. These are seen in photographic
emulsions and in cloud chambers. The prongs are due to ionizing
^articles, protons, albhas, or larger fragments. Neutrons leave
no trace. The quantitative interpretation is rudimentary. Star
production car be interpreted roughly as evaporation of particles
from a very high temperature nucleus. M. Goldberger*- has added
to the evaporation model consideration of the situation when a
very high energy nucleon, of energy around 100 Mev, has first
entered the nucleus. At this tine, when its wavelengh is very
email and its energy is still undistributed
among the other nucleons, it has collisions
which knock other nucleons out of the nuc-
leus immediately. Soon, however, the energy
not curried off by these quickly escaping
nucleons is distributed in the form of
statistical excitation of the nucleus.
Then the escape of nucleons is described
by the slower arocess of evaporation. Goldberger calculates that
about 1/2 of the initial 100 Mev leaves immediately t and about
1/2 is trapped and produces evaporation.
b) Deuteron st r 1 y, ping ■ ** Deuterons, given an energy of 200
Mev In the Berhelev 184" cyclotron, impinge on a target Inside
the cyclotron, and" a beam of very high energy neutrons emerges.
(FI&. VII I. 14 J The interpretation is that the proton of the
loosely bound deuteron hits and is caught by a nucleus, and the
neutron flies on. Ideally, the neutron would have about 1/2 the
energy of the deuteron. However the neutron has a velocity with
resnect to the center of mass of the deuteron. After the proton
is removed, the neutron's total velocity vector is the vector sum
of the velocity of the center of ma s s of the d eu t e r on and th e v el -
ocitv of the neutron relative to the center of mass at the Instant
the proton is removed. Therefore the neutrons emerge with a
spread in energy and in angle .
StaP
* M.L. Goldberger, Phys.Rev . Jh 1269 (1948)
** Helrr.olz, McMillan and Sewell, Phy s . Rev . jS 1003 (1947)
178
Very High Energy Phenomena
Ch. YIII
>m
tyclotrcn
tjeuteron ^am
CYe.UtVW (of at on
oeam
FIG. VIII. 14
FIG. VI II. 15
e;) b s e rv ocl exchange 1 ri p- n 3 c a 1 1 e r i n g . Wh e r. a be am of
350 Mey protona hits a scatterer placed, inside the Berkeley cyclo-
tron, FIG. VI II. 15 , high energy neutrons (as well as protons)
emerge. This shov.'s that in many collisions the neutron and
proton exchange roles: 3ccttie rer
^ ^__ . ^- R^jftyon
A similar effect has been observed in ft-.-rj scattering.'*
Problem: Compute the distribution in energy and angle of scat-
tering of neutrons re suiting from the stripping of fast deuter-
ons , using the simple mod~l described above.
(This calculation Is done in the first part of Berber 1 s paper
on the theory of stripping of deuterons, Phy a ■ Rev . 7_2 1007
(l94y). Although the most probable neutron energy is l/2 that
of the deuteron, the energy spread, is larger than the binding
energy of the deuteron by a factor of (l/2 E d /EE(d) ) /i «{ 100/2 y z
ar 7. This may seem like violation of cor.sen.rat ion of energy,
Houever, the velocity of the neutron relative to the laboratory
system = '^^_ elx ^ t + ^ rel _ c i eu t. > and - t - le elle - T CJ °f the neutron
In the lab. system thereforeis
^M(aij 1 + fe.^ifed^O^ + Zarj^^.fad-t)
:ere the last term, is the marl
contribution to the snread,
For a su mm a ry of very hi gh e ne r f-y phenomena, s e i
Chew and Moyer, Are. J. Physics IS 1S5-1J5, Ig 17, 22 20 5-
*This is discussed further under "Exchange Forces" in Ch. VI, p. 122